The proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra difference equations with delays

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Vol. 19 (2018), No. 1, pp. 95–109 DOI: 10.18514/MMN.2018.2335

BIBO STABILITY OF DISCRETE CONTROL SYSTEMS WITH SEVERAL TIME DELAYS

ESSAM AWWAD, ISTV ´AN GY ˝ORI, AND FERENC HARTUNG Received 23 May, 2017

Abstract. This paper investigates the bounded input bounded output (BIBO) stability in a class of control system of nonlinear difference equations with several time delays. The proofs are based on our studies on the boundedness of the solutions of a general class of nonlinear Volterra difference equations with delays.

2010Mathematics Subject Classification: 39A30; 93C55

Keywords: boundedness, Volterra difference equations, bounded input bounded output (BIBO) stability, difference equations with delays

1. INTRODUCTION

Time delays play an important role in control systems, since a delay naturally appears when a system wants to measure or react to information. Stability or stabilization of a system is one of the central question which is investigated in control theory [10–12]. Because of its simplicity, the bounded input bounded output (BIBO) stability of control systems is widely investigated. The sufficient conditions for BIBO stability of control systems without delays are obtained in [18,19] by using Liapunov function techniques. More recently many researchers have focused their interest on the BIBO stability of nonlinear discrete and continuous feedback control systems with or without delays [1,2,5–9,13–15,17].

In this paper we consider a class of discrete control systems with multiple time delays. We search for delayed feedback controls such that the corresponding closed loop system be BIBO stable. We rewrite the closed loop system as an equivalent nonlinear Volterra difference equation (VDE) with delays. The BIBO stability results are based on our theorem which formulate sufficient conditions for the boundedness of the solutions of delayed VDEs. The results presented in this manuscript extend the methods introduced in [1] for nonlinear differential equations with a single delay and boundedness of ordinary VDEs presented in [3].

The structure of the manuscript is the following. Section2 contains the precise problem statement, the definitions of BIBO stability and local BIBO stability, and we

c 2018 Miskolc University Press

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rewrite our closed loop control equation as an equivalent VDE. Section3formulates sufficient conditions for the boundedness of a general class of nonlinear VDEs with multiple delays. Section 4 contains our BIBO stability results for cases when the nonlinearity has a sub-linear, linear or super-linar estimates.

In the rest of this section we introduce some notations which will be used through- out this paper. R, R_C, R^d and R^d^d denote the set of real numbers, nonnegative real numbers, d-dimensional real column vectors and dd-dimensional real matrices, respectively. The maximum norm onR^d is denoted byk k, i.e.,kxk WD max1idjxij, where xD.x1; : : : ; xd/^T. The matrix norm onR^d^d generated by the maximum vector norm will be denoted byk k, as well. LetZ_CandNbe the set of nonnegative and positive integers, respectively. L¹.Z_C;R^d/will denote the set of bounded sequencesrWZ_C!R^d with normkrk1WDsup_n₂_Z_Ckr.n/k. Let > 0 be a fixed integer,S.Œ ; 0;R^d/denotes the set of finite sequences

n

W f ; C1; : : : ; 0g !R^d o

andk kWD max

n0k .n/k. For a given sequencexand an integernthe forward difference operator is defined byx.n/WDx.nC1/ x.n/.

2. PROBLEM STATEMENT

In this paper we consider the nonlinear discrete control system with several delays x.n/Dg.n; x.n 1.n//; : : : ; x.n _`.n///Cu.n/; n2Z_C;

y.n/DC x.n/; n2Z_C: (2.1)

Herex.n/2R^d is the state vector,u.n/2R^d is the input vector andy.n/2R^d¹ is the output vector of the system (2.1),C 2R^d¹^d is a constant matrix,i W Z_C! Z_C, i D1; : : : ; ` are bounded delay functions, and the nonlinear functiongWZ_C R^d: : :R^d

„ ƒ‚ …

`

!R^d satisfies

kg.n; x^.1/; : : : ; x^.`//k b.n/'

max

1m`kx^.m/k

; n2Z_C; x^.1/; : : : ; x^.`/2R^d; (2.2) where b.n/ > 0for all n2Z_C, and' W R_C!R_C is a monotone nondecreasing mapping.

Our general problem (2.1) satisfying condition (2.2) includes, e.g., linear control systems

x.n/DA1.n/x.n 1.n//C CA`.n/x.n `.n//Cu.n/; n2Z_C;

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and nonlinear control systems of the form xi.n/D

d

X

jD1

aij.n/x_j^p.n j.n//Cui.n/; n2Z_C; iD1; : : : ; d;

wherex.n/D.x1.n/; : : : ; xd.n//^T, u.n/D.u1.n/; : : : ; ud.n//^T,p > 0; or a poly- nomial difference system

xi.n/D

`

X

jD1

aij.n/x₁^q^ij1.n j.n// x_d^q^ijd.n j.n//Cui.n/

forn2Z_C; iD1; : : : ; d, whereq_{ij k}2R_Cfori; kD1; : : : ; d andj D1; : : : ; `; or the scalar nonautonomous control system of the form

x.n/D a.n/x^p.n 1.n//

b.n/Cx^q.n 2.n//Cu.n/; n2Z_C;

wherep; q > 0. In all the above cases assumption (2.2) holds under natural conditions with'.t /Dt^p with somep > 0.

We assume that the uncontrolled system, i.e., (2.1) with u0 has unbounded solutions. Our goal is to find a positive diagonal matrixD and a positive integerk such that the delayed feedback law of the form

u.n/D Dx.n k/Cr.n/ (2.3)

guarantees that the closed-loop delayed system

x.n/Dg.n; x.n 1.n//; : : : ; x.n `.n/// Dx.n k/Cr.n/; n2Z_C; y.n/DC x.n/; n2Z_C;

x.n/D .n/; n2 f ; C1; : : : ; 0g

(2.4) is BIBO stable. Herer.n/ is the reference input,DDdiag.1; : : : ; d/, i > 0for i D1; : : : ; d, 2S.Œ ; 0;R^d/ is the initial sequence associated to the equation where

WDmax

1maxj`kjk1; k

; (2.5)

The assumed diagonal form of the feedback law (2.3) is one of the simplest possible choice. In its implementation it is important to know how large delay can be. In The- orem2and3we give sufficient conditions on how to select the feedback gainDand the time delaykto guarantee the boundedness of the solutions. Our conditions (see (4.1) and (4.13) below) show that the larger the delay the smaller gain can guarantee the boundedness of the solution.

Following [16], we introduce the next definition of BIBO stability.

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Definition 1. The closed loop system (2.4) is said to be BIBO stable if there exist positive constants1and2D2.k k/such that every solution of the system (2.4) satisfies

ky.n/k 1krk1C2; n2Z_C for every reference inputr2L¹.Z_C;R^d/.

Later we need the notion of local BIBO stability (see similar definition in [1] for the continuous case).

Definition 2. The closed loop system (2.4) is said to be locally BIBO stable if there exist positive constantsı1,ı2and satisfying

ky.n/k ; n2Z_C provided thatk k< ı1andkrk1< ı2.

Our approach is the following. We associate the linear system

´.n/D D´.n k/; n2Z_C (2.6)

with the constant delayk2Nand the initial condition

´.n/D .n/; kn0 (2.7)

to (2.4). Then the state equation in (2.4) can be considered as the nonlinear perturbation of (2.6), and by the variation of constants formula (see, e.g., Lemma 4 in [4]) we get

x.n/D´.n/C

n 1

X

jD0

W .n j 1/ Œg.j; x.j 1.j //; : : : ; x.j _`.j ///Cr.j /

(2.8) forn2Z_C, where´.n/is the solution of (2.6)-(2.7) andW is the fundamental matrix solution of (2.6), i.e., the solution of the IVP

W .n/D DW .n k/; n2Z_C; (2.9)

W .n/D

0; kn 1, I; nD0.

Here I2R^d^d is the identity matrix and02R^d^d is the zero matrix. SinceDis a diagonal matrix, it is easy to see thatW .n/is a diagonal matrix too for alln2Z_C.

We can rewrite the equation (2.8) as a VDE x.nC1/D´.nC1/C

n

X

jD0

W .n j /g.j; x.j 1.j //; : : : ; x.j _`.j ///

C

n

X

jD0

W .n j /r.j /; n2Z_C;

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and so it is equivalent to x.nC1/D

n

X

jD0

f .n; j; x.j 1.j //; : : : ; x.j _`.j ///Ch.n/; n2Z_C; (2.10) where

h.n/WD´.nC1/C

n

X

jD0

W .n j /r.j /; (2.11)

and

f .n; j; x^.1/; : : : ; x^.`//WDW .n j /g.j; x^.1/; : : : ; x^.`// (2.12) for 0j n; x^{.i /}2R^d; 1i`. The equation (2.10) is a nonlinear VDE with several delay functions.

3. BOUNDEDNESS OF THE SOLUTIONS OF VDES WITH DELAYS

In this section we give a general result for the boundedness of the solutions of nonlinear VDEs with multiple delays which is a natural extensions of the results presented in [3] for nonlinear VDEs without delays.

We consider the nonlinear VDE with several delays x.nC1/D

n

X

jD0

f .n; j; x.j 1.j //; : : : ; x.j _`.j ///Ch.n/; n2Z_C; (3.1) with the associated initial condition

x.n/D .n/; n0; (3.2)

where is a positive integer constant. We assume the following conditions.

(B1) For any fixed0j nandj; n2Z_C f .n; j;; : : : ;/WR^d: : :R^d

„ ƒ‚ …

`

!R^d:

(B2) For any0jnand1i d there exists anai.n; j /2R_Csuch that jfi.n; j; x^.1/; : : : ; x^.`//j ai.n; j /'

max

1m`kx^.m/k

(3.3) holds forx^.1/; : : : ; x^.`/2R^d with a monotone non-decreasing mapping 'W R_C!R_C, wheref D.f1; : : : ; f_d/^T.

(B3) h.n/D.h1.n/; : : : ; h_d.n//^T 2R^dforn2Z_C.

(B4) i W Z_C!Z_Csatisfiesji.n/j forn2Z_CandiD1; : : : ; `.

(B5) 2S.Œ ; 0;R^d/.

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Clearly, problem (3.1)-(3.2) has a unique solution under the above conditions. The next result formulates sufficient conditions implying the boundedness of the solutions.

Theorem 1. Let be fixed, (B1)-(B5) are satisfied and letx.nI /be the solution of (3.1)-(3.2). Suppose there exist N 2Z_C, 2R_C andv such that for i D 1; : : : ; d

N

X

jD0

ai.N; j /'./C jhi.N /j v; (3.4)

N

X

jD0

ai.n; j /'./C

n

X

jDNC1

ai.n; j /'.v/C jhi.n/j v; nNC1 (3.5) and

jjx.nI /jj ; n2 f ; : : : ; Ng: (3.6) Then the solution is bounded byv, i.e.

jjx.nI /jj v; n : (3.7)

Proof. Consider the solution x.n/ Dx.nI /, n 2Z_C of (3.1) with the initial condition (3.2), and letandN be such that (3.6) holds. Then, by using (B2), (3.4), (3.6) and the monotonicity of', we have foriD1; : : : ; d

jxi.NC1/j

N

X

jD0

jfi.N; j; x.j 1.j //; : : : ; x.j _`.j ///j C jhi.N /j

N

X

jD0

ai.N; j /'. max

mNkx.m/k/C jhi.N /j

N

X

jD0

a_i.N; j /'./C jh_i.N /j v;

Thereforekx.NC1/k v, so (3.7) holds fornDNC1.

Now we show that (3.7) holds for anynNC1. Assume, for the sake of contra- diction, that there existsn0NC1andi02 f1; : : : ; dgsuch that

jxi0.n0C1/j D jxi0.n0C1I /j> v; (3.8) and

jxi.n/j D jxi.nI /j v; NC1nn0; iD1; : : : ; d: (3.9)

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Hence, from equation (3.1), we get jxi0.n0C1/j

N

X

jD0

jfi0.n0; j; x.j 1.j //; : : : ; x.j `.j ///j

C

n0

X

jDNC1

jfi0.n0; j; x.j 1.j //; : : : ; x.j _`.j ///j C jhi0.n0/j

N

X

jD0

ai0.n0; j /'. max

mNkx.m/k/ C

n0

X

jDNC1

ai₀.n0; j /'. max

mjkx.m/k/C jhi₀.n0/j: Since'is a monotone non-decreasing mapping, (3.5), (3.6) and (3.9) yield

jxi₀.n0C1/j

N

X

jD0

ai₀.n0; j /'./C

n0

X

jDNC1

ai₀.n0; j /'.v/C jhi₀.n0/j v:

This contradicts to our hypothesis (3.8), so inequality (3.7) holds.

4. MAIN RESULTS

Our main goal in this section is to formulate sufficient conditions which grantee the BIBO stability of the closed loop system (2.4). We will assume that function'in (2.2) is a power function. Our first result is given for the case whengin (2.2) has a sub-linear estimate, i.e., when'.t /Dt^p;with0 < p < 1in (2.2).

Theorem 2. LetgWR^d !R^d be a function which satisfies inequality(2.2) with '.t / Dt^p; 0 < p < 1, t 0. The feedback control system (2.4) with D D diag.1; : : : ; _d/andk2Nis BIBO stable if

kbk1WD sup

n2Z_C

b.n/ <1 and 0 < i < 2cos k

2kC1; iD1; : : : ; d (4.1) hold.

Proof. Let D. 1; : : : ; d/^T 2S.Œ ; 0;R^d/, and´.n/D.´1.n/; : : : ; ´d.n//^T be the solution of the IVP (2.6)-(2.7). Then, foriD1; : : : ; d; ´i is the solution of the IVP

´i.n/D i´i.n k/; n2Z_C (4.2)

with initial condition

´i.n/D i.n/; kn0: (4.3)

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It is known (see, e.g., [4]) that condition (4.1) yields that there exists a positive con- stantM and2.0; 1/such that

j´i.n/j Mk kⁿ; n2Z_C; i D1; : : : ; d; (4.4) wherek kWDmax j0k .j /k. Hence every solution of (4.2) tends to zero as n! 1, and

k´k1WD sup

n2ZC

k´.n/k Mk k <1: (4.5) LetW .n/Ddiag.w1.n/; : : : ; w_d.n// be the solution of (2.9). Relation (4.4) yields limn!1wi.n/D0foriD1; : : : ; d, and

WD max

0id 1

X

nD0

jwi.n/j<1: (4.6) From (2.8), for alln2Z_CandiD1; : : : ; d, we have

xi.nC1/D´i.nC1/

C

n

X

jD0

wi.n j / Œgi.j; x.j 1.j //; : : : ; x.j _`.j ///Cri.j / ; (4.7) wherex.n/D.x1.n/; : : : ; x_d.n//^T,gD.g1; : : : ; g_d/^T andrD.r1; : : : ; r_d/^T. There- fore (2.11) and (2.12) imply

fi.n; j; x^.1/; : : : ; x^.`//Dwi.n j /gi.j; x^.1/; : : : ; x^.`// and

hi.n/D´i.nC1/C

n

X

jD0

wi.n j /ri.j /:

Hence, by (2.2),

jf_i.n; j; x^.1/; : : : ; x^.`//j jw_i.n j /j jg_i.j; x^.1/; : : : ; x^.`//j jwi.n j /jb.j /'

maxm`kx^.m/k

; so the conditions (B1)-(B5) hold withai.n; j /WD jwi.n j /jb.j /,0j n.

By (4.1), (4.5), (4.6) and the definition of the infinity norm, we obtain WD max

1id sup

n2ZC

jhi.n/j

max

1id sup

n2Z_Cj´i.n/j C max

1id sup

n2Z_C n

X

jD0

jwi.n j /jkr.j /k

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sup

n2Z_Ck´.n/k C krk1 max

1id 1

X

jD0

jwi.j /j (4.8)

D k´k1Ckrk1<1: (4.9)

By conditions (4.1) and (4.6) we get

˛W D max

1id sup

n2Z_C n

X

jD0

ai.n; j /

D max

1id sup

n2Z_C n

X

jD0

jwi.n j /jb.j /

kbk1 max

1id 1

X

jD0

jwi.j /j

Dkbk1<1: (4.10)

Now we show that the inequalities (3.4) and (3.5) are satisfied with '.t /Dt^p; t 0; N D0; WD k k WD max

n0kx.n/k (4.11) and

vWDmax

2.kbk1k k^p C k´k1Ckrk1/; .2kbk1/¹¹^p;k k

: (4.12) By using (4.9) and (4.10), it is clear that foriD1; : : : ; d

ai.0; 0/k k^p C jhi.0/j kbk1k k^p C k´k1Ckrk1v;

therefore (3.4) holds with (4.11) and (4.12). We have v.2kbk1/¹¹^p, and so (4.10) and the definition of˛yield

v^{p 1}˛ ˛

2kbk1 1 2:

Similarly, usingv2.kbk1k k^pCk´k1Ckrk1/and the inequalities (4.9) and (4.10), we obtain

1

v.˛k k^p C / ˛k k^p C

2.kbk1k k^p C k´k1Ckrk1/ 1 2: Thus

v^{p 1}˛C1

v.˛k k^p C /1;

hence for alln1, we have foriD1; : : : ; d ai.n; 0/'.jj jj/C

n

X

jD1

ai.n; j /'.v/C jhi.n/j ˛k k^p C˛v^pC v;

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consequently, (3.6) holds with (4.11) and (4.12). Then all the conditions of Theorem 1are satisfied, therefore the solutionxof the closed loop system (2.4) is bounded by vforn , i.e.,

kx.n/k vDmax

2.kbk1k k^p C k´k1Ckrk1/; .2kbk1/¹¹^p;k k 2krk1Cmax

2.kbk1k k^p C k´k1/; .2kbk1/¹¹^p;k k

forn . Then

ky.n/k kCkkx.n/k 1krk1C2; n2Z_C; where1WD2kCkand

2WD kCkmax

2.kbk1k k^p C k´k1/; .2kbk1/¹¹^p;k k

:

Hence, by Definition1, the closed loop system (2.4) is BIBO stable.

It is easy to see that forkD1the last inequality of (4.1) gives the upper bound i< 1, and ask! 1, the upper bound ofi in condition (4.1) tends monotonically to 0. Therefore large delay allows only small gain in the control law.

In the following theorem a sufficient condition is given for the BIBO stability in the case of a linear estimate of the functiong.

Theorem 3. LetgWR^d!R^d be a continuous function which satisfies inequality (2.2) with'.t /Dt; t 0. The closed loop system (2.4) withDDdiag.1; : : : ; _d/ andk2Nis BIBO stable if

kbk1< 1

and 0 < i < 2cos k

2kC1; iD1; : : : ; d (4.13) hold, whereis defined by (4.6).

Proof. As in the proof of Theorem2, we rewrite (2.4) in the form of (4.7), and define the functionsfi,ai andhiforiD1; : : : ; d. Then the conditions (B1)-(B5) are satisfied.

Next we show that the inequalities (3.4) and (3.5) are satisfied with '.t /Dt; N D0; WD k k and vWDmax

krk1CMk k 1 kbk1

;k k

; (4.14) where the positive constantM is defined in (4.4), k k WD sup

n0kx.n/k. Since ai.n; j /WDwi.n j /b.j /,0j n, we have

n

X

jD0

ai.n; j /D

n

X

jD0

jwi.n j /jb.j /

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max

1id sup

n2Z_C n

X

jD0

jwi.n j /jb.j /

kbk1 max

1iq 1

X

nD0

jwi.n/j

Dkbk1 (4.15)

< 1: (4.16)

By (4.4), (4.8), (4.13), (4.14) and (4.16), we have forn2Z_C,iD1; : : : ; d vkrk1CMk k

1 kbk1

krk1

n

X

jD0

jwi.n j /j C j´i.nC1/j

1

n

X

jD0

jwi.n j /jb.j /

n

X

jD0

jwi.n j /jjri.j /j C j´i.nC1/j

1

n

X

jD0

jwi.n j /jb.j / :

Sincehi.n/DPn

jD0wi.n j /r.j /C´i.nC1/, it follows v jhi.n/j

1

n

X

jD0

jwi.n j /jb.j /

; n2Z_C:

Therefore v

n

X

jD0

jwi.n j /jb.j /C jhi.n/j v; n2Z_C; iD1; : : : ; d:

Hence the above inequality andv k k yield fornD0andiD1; : : : ; d ai.0; 0/k kC jhi.0/j vjwi.0/jb.0/C jhi.0/j v;

and so (3.4) is satisfied with (4.14). Similarly, forn2NandiD1; : : : ; d vjwi.n/jb.0/Cv

n

X

jD1

jwi.n j /jb.j /C jhi.n/j v:

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Therefore

ai.n; 0/k kCv

n

X

jD1

ai.n; j /C jhi.n/j v; n2N; iD1; : : : ; d;

consequently, (3.5) is satisfied with (4.14). Then all the conditions of Theorem1hold with with (4.14), therefore the solutionxof the closed loop system (2.4) is bounded byv, i.e.,

kx.n/k v; n2Z_C: Hence

ky.n/k kCkkx.n/k kCkv D kCkmax

krk1CMk k 1 kbk1

;k k

1krk1C2; where

1WD kCk 1 kbk1

and 2WD kCkmax

Mk k

1 kbk1

;k k

:

Then, by Definition1, the feedback control system (2.4) is BIBO stable.

Corollary 1. LetgWR_CR^d!R^d be a continuous function which satisfies inequality (2.2) with '.t / D t, t 0. The closed loop system (2.4) with D D diag.1; : : : ; _d/andk2Nis BIBO stable if

kbk1< i k^k

.kC1/^k^C¹; iD1; : : : ; d (4.17) hold.

Proof. Under our condition (4.17) and from Lemma 4 in [4] we get that the fundamental solutionwi of (4.2)-(4.3) is positive and

1

X

jD0

wi.j /D 1 i

; i D1; : : : ; d:

Therefore

Dmax 1

1

; : : : ; 1 _d

;

and hence kbk1 < 1. The proof is similar to the proof of Theorem 3 and it is

omitted.

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In the next theorem it is shown that in the super-linear case there exist positive diagonal gainDand positive delayksuch that the solutions of the closed loop system are bounded for small initial functions and small reference inputs, i.e., the system is locally BIBO stable.

Theorem 4. LetgWR^d!R^d be a continuous function which satisfies inequality (2.2) with'.t /Dt^p; p > 1,t0. Then the solutionxof the feedback control system (2.4) is locally BIBO stable if (4.1) holds.

Proof. Suppose 1; : : : ; _d are fixed satisfying (4.1),DDdiag.1; : : : ; _d/, and let´be the solution of the IVP (2.6)-(2.7), andbe defined by (4.6). Letk kı1

andkrk1ı2, whereı1,ı2will be specified later. From (4.5) and (4.9) we have k´k1Mk kM ı1

and

WD max

1id sup

n2Z_C

jhi.n/j k´k1Ckrk1M ı1Cı2<1; and from (4.1) and (4.15) it follows

˛WD max

1id sup

n2Z_C n

X

jD0

ai.n; j /kbk1<1:

Sincep > 1andandkbk1 are positive and finite, we select the positive constants ı1andı2so that

˛ı^p₁CM ı1Cı2 1 2

1 2kbk1

_p¹₁

and 0 < ı1 1

2kbk1

_p¹₁

(4.18) hold.

Next we show that the inequalities (3.4) and (3.5) are satisfied with '.t /Dt^p; N D0; WD k k and vWD

1 2kbk1

_p¹₁

: (4.19) We note that the definitions of,ı1and the second part of (4.18) yieldv. Using the definition ofv,p > 1and (4.18) we get

v 1 2

1 2kbk1

_p¹₁

˛ı₁^pCM ı1Cı2˛k k^pCai.0; 0/k k^pC jhi.0/j fori D1; : : : ; d, hence the condition (3.4) holds with (4.19).

Similarly, the definition ofv,p > 1and (4.18) yield v ˛v^p v kbk1v^pD1

2 1

2kbk1

_p¹₁

˛ı₁^pCM ı1Cı2˛k k^p C:

(14)

Then the definitions of˛and imply ai.n; 0/k k^p C

n

X

jD1

ai.n; j /v^pC jhi.n/j ˛k k^p C˛v^pC v; n2N;

therefore the condition (3.5) holds with (4.19).

Therefore the conditions of Theorem1are satisfied with (4.19), so the solution of the closed loop system (2.4) is bounded byv, i.e.,

kx.n/k< vD 1

pkbk1

_p¹₁

; n2Z_C: Hence

ky.n/k kCkkx.n/k ; n2Z_C; where

WD kCk 1

pkbk1

_p¹₁ :

By Definition2the closed loop system (2.4) is locally BIBO stable.

ACKNOWLEDGEMENT

This research was partially supported by by the Hungarian National Foundation for Scientific Research Grant No. K120186.

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Authors’ addresses

Essam Awwad

Department of Mathematics, Faculty of Science, Benha University, Egypt E-mail address:esam mh@yahoo.com

Istv´an Gy˝ori

Department of Mathematics, University of Pannonia, Hungary E-mail address:gyori@almos.uni-panon.hu

Ferenc Hartung

Department of Mathematics, University of Pannonia, Hungary E-mail address:hartung.ferenc@uni-pannon.hu